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Composing Functions

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Description:

- To understand how to compose functions
- To know how determine the domain of function compositions
- To see how new functions may be defined by composing given functions
- To see how some functions can be decomposed in terms of simpler functions

The concept of function composition is first motivated by comparing it to basic arithmetic operations. Next the composition of a function is defined and numerous examples are provided wherein functions are composed. The effect of function composition on domain and range is mentioned along with a demonstrative example, and the associativity property is discussed. Finally function decomposition is introduced by example.

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Tutorial

Prelude

Background

In order to benefit from this lesson you ought to have a firm understanding of functions, domain, and range, and you should be familiar with rational functions since each will be used in the examples. You will also need to know how to solve inequalities, these too will appear in the examples.

Direction

As abstract concepts go, we are amazingly comfortable with numbers. We experience absolutely no discomfort when we add, subtract, multiply and even divide numbers with one another. We can even construct mathematical functions that manipulate numbers and perform a wide variety of exotic computations on them. Yes, numbers our our playthings.

What we're going to learn is that functions too can be just as malleable as numbers. We will learn to think of a function as just another abstraction. And as abstraction entities all their own, we will start to see functions as simple things that can be operated upon as easily as we operate on numbers. This lesson introduces the operation known as function composition. Just as we can add or subtract two numbers, so too can compose two functions.

Compositions

What's it about?

Function composition is a way to take two smaller functions and hook them together to make one larger function. The advantage is that by composing functions you reduce the number of parts to keep track of in your mathematical work - simpler is better, and one is simpler than two.

How does it work?

What function composition actually does is pretty simple: if you have two functions f(x) and g(x), and you want to compose them, written g º f (x), then you just use the output of one function as the input of the other. For example, if f(x) = x+1, and g(x) = x2, then g º f (x) = (x+1)2. Why? Because the output of f(x) is x+1 which we will use as the input of g(x). This means that wherever we see an x variable within the expression defining g(x), we replace it with x+1, hence (x+1)2

Another often used notation for the same thing is g(f(x)). Sometimes people prefer this notation when they need to actually find the formula for the composed functions because it makes the process more intuitive. Here is the example from the last paragraph: g(f(x))= g(x+1) = (x+1)2.

An Example With A Moral

So in general g º f (x) is not the same thing as f º g (x). There are some cases, however, where the two are equivalent. For example, if h(x) = x2 and k(x) = x3, then

h(k(x)) = h(x3) = (x3)2 = x6 = (x2)3 = k(x2) = k(h(x)).

Such situations are, however, exception to the rule.

Composition and Function Domain

Using the same f(x) and g(x) from the example above, we now find the domain of their compositions.

New From Old

Often, it is convenient to give the composition of two functions a name of its own. The following example demonstrates that the composition of two functions is just another function.

Associativity Property

Composing functions is an associative property. Associativity means that when we are given three functions, f(x), g(x), and h(x), then composing g(x) with f(x) and then composing h(x) with the result is the same thing as composing h(x) with g(x) and then composing result with f(x). In other words

h º (g º f) = (h º g) º f,

where the parentheses group terms by which composition happens first.

In the following example we will use three functions:

Each of these functions is color coded. At each step, the expression on either side of the equal sign changes, and the colors show which function is responsible for that change. In the end, the same expression is found on both sides of the equal sign.

This shows that function composition is associative.

Function Decomposition

A Certain Symmetry

So far we have been using the concept of function composition to build larger functions out of smaller ones. Looking at some of those compositions above, we see that we could also take a larger function and break it down into smaller ones by using the reverse process. If you're given a function h(x) and you see that it is the composition of two simpler functions g(x) and f(x), then you have just found a decomposition of h(x). Here are some examples of decomposing functions into simpler ones.